[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:03.50,0:00:07.55,Default,,0000,0000,0000,,In the last segment, I showed you how\Nreturn maps work,
Dialogue: 0,0:00:07.58,0:00:10.50,Default,,0000,0000,0000,,how to go back and forth between \Nthem and the time domain,
Dialogue: 0,0:00:10.54,0:00:14.43,Default,,0000,0000,0000,,and how they help you understand the\Ndynamics, as well as
Dialogue: 0,0:00:14.47,0:00:19.20,Default,,0000,0000,0000,,how to understand bifurcations in the\Ndynamics as the parameter value changes.
Dialogue: 0,0:00:19.80,0:00:24.74,Default,,0000,0000,0000,,I finished up with a 3rd representation,\Nthe bifurcation diagram.
Dialogue: 0,0:00:25.48,0:00:28.76,Default,,0000,0000,0000,,Here's a bifurcation diagram of \Nthe logistic map.
Dialogue: 0,0:00:29.24,0:00:32.81,Default,,0000,0000,0000,,On the vertical axis is a set of iterates\Nof the logistic map,
Dialogue: 0,0:00:33.19,0:00:36.83,Default,,0000,0000,0000,,at some parameter value R, which is \Ngraphed on the horizontal axis.
Dialogue: 0,0:00:37.15,0:00:40.80,Default,,0000,0000,0000,,Just to remind you of the correspondence\Nbetween this kind of plot
Dialogue: 0,0:00:40.88,0:00:44.60,Default,,0000,0000,0000,,and the time domain, and the return map,\NI'm gonna draw a few pictures.
Dialogue: 0,0:00:44.91,0:00:48.70,Default,,0000,0000,0000,,Here's a time domain plot of an orbit of\Nthe logistic map
Dialogue: 0,0:00:48.80,0:00:52.96,Default,,0000,0000,0000,,at a low value of the parameter R\Nthat is converging to a fixed point.
Dialogue: 0,0:00:53.51,0:00:57.04,Default,,0000,0000,0000,,On a return map, this orbit would look\Nlike this.
Dialogue: 0,0:00:58.06,0:01:01.57,Default,,0000,0000,0000,,To construct a bifurcation diagram,\Nyou remove the transient;
Dialogue: 0,0:01:01.60,0:01:04.82,Default,,0000,0000,0000,,that is, you iterate a bunch of times,\Nand throw those points away,
Dialogue: 0,0:01:04.89,0:01:08.08,Default,,0000,0000,0000,,and then you iterate a bunch more times,\Nand you plot those points
Dialogue: 0,0:01:08.14,0:01:11.42,Default,,0000,0000,0000,,as if you were looking at that top plot \Nedge-on from the side.
Dialogue: 0,0:01:12.10,0:01:15.45,Default,,0000,0000,0000,,In this case, those points would all fall\Non top of each other, there.
Dialogue: 0,0:01:16.15,0:01:18.84,Default,,0000,0000,0000,,So again, each vertical slice of the \Nbifrucation diagram
Dialogue: 0,0:01:18.84,0:01:22.46,Default,,0000,0000,0000,,is one time-domain plot like this, with \Nthe transient removed,
Dialogue: 0,0:01:22.73,0:01:23.92,Default,,0000,0000,0000,,viewed from the side.
Dialogue: 0,0:01:24.30,0:01:27.89,Default,,0000,0000,0000,,If we turn R up a little bit, the time-\Ndomain plot will look like this,
Dialogue: 0,0:01:28.34,0:01:30.05,Default,,0000,0000,0000,,the return map will look like this,
Dialogue: 0,0:01:30.56,0:01:34.12,Default,,0000,0000,0000,,and the point on the bifurcation diagram\Nwill look like 2 dots.
Dialogue: 0,0:01:34.20,0:01:37.60,Default,,0000,0000,0000,,Again, three different representations\Nbring out three different things:
Dialogue: 0,0:01:37.60,0:01:39.35,Default,,0000,0000,0000,,the time-domain plot on the top left
Dialogue: 0,0:01:39.35,0:01:41.67,Default,,0000,0000,0000,,brings out the overall behavior of \Nthe iterates;
Dialogue: 0,0:01:41.67,0:01:44.99,Default,,0000,0000,0000,,the return map on the lower left\Nbrings out the geometry of why
Dialogue: 0,0:01:44.99,0:01:48.97,Default,,0000,0000,0000,,the iterates go where they go, & also the \Ncorrelation between successive iterates;
Dialogue: 0,0:01:49.01,0:01:53.27,Default,,0000,0000,0000,,the bifurcation plot brings out what\Nchanges about the asymptotic behavior
Dialogue: 0,0:01:53.27,0:01:56.86,Default,,0000,0000,0000,,of the trajectory as R changes, \Nincluding bifurcations.
Dialogue: 0,0:01:57.45,0:02:02.08,Default,,0000,0000,0000,,Now if you repeat the procedure that we\Njust went through at a much finer grain,
Dialogue: 0,0:02:02.22,0:02:06.23,Default,,0000,0000,0000,,but using a computer instead of tablet \Nand a stylus, what you'll see is this.
Dialogue: 0,0:02:06.38,0:02:09.55,Default,,0000,0000,0000,,There's actually one more step in there\Nwhich we'll circle back to
Dialogue: 0,0:02:09.57,0:02:11.01,Default,,0000,0000,0000,,at the end of this segment.
Dialogue: 0,0:02:11.17,0:02:13.93,Default,,0000,0000,0000,,Now, you can see all sorts of structure\Nin this plot.
Dialogue: 0,0:02:14.02,0:02:16.26,Default,,0000,0000,0000,,That's the main focus of this segment.
Dialogue: 0,0:02:16.49,0:02:20.09,Default,,0000,0000,0000,,First of all, you see the fixed point\Ncoming along for low R, here,
Dialogue: 0,0:02:20.35,0:02:23.14,Default,,0000,0000,0000,,and then bifurcating into a 2-cycle\Nright here,
Dialogue: 0,0:02:23.32,0:02:26.52,Default,,0000,0000,0000,,bifurcating into a 4-cycle right here,\Nand then eventually,
Dialogue: 0,0:02:26.71,0:02:30.89,Default,,0000,0000,0000,,getting into a chaotic regime. That's \Nwhat this gray banded behavior is.
Dialogue: 0,0:02:31.24,0:02:34.94,Default,,0000,0000,0000,,That's what this right-hand plot would\Nlook like if you looked at it edge-on
Dialogue: 0,0:02:35.04,0:02:37.35,Default,,0000,0000,0000,,from the right-hand side of the screen.
Dialogue: 0,0:02:37.46,0:02:41.20,Default,,0000,0000,0000,,Within the chaotic regimes, you also see\Nthese "veils":
Dialogue: 0,0:02:41.57,0:02:44.44,Default,,0000,0000,0000,,areas where the attractor is darker\Nthan in other areas.
Dialogue: 0,0:02:44.85,0:02:49.30,Default,,0000,0000,0000,,Those veils are related to what are called\N"unstable periodic orbits",
Dialogue: 0,0:02:49.58,0:02:51.85,Default,,0000,0000,0000,,and we'll talk more about them later.
Dialogue: 0,0:02:51.89,0:02:54.40,Default,,0000,0000,0000,,As we've seen, there's this bifurcation \Nsequence
Dialogue: 0,0:02:54.53,0:02:59.16,Default,,0000,0000,0000,,from a fixed point, to a 2-cycle, to a 4-\Ncycle, to an 8-cycle, & so on & so forth.
Dialogue: 0,0:02:59.46,0:03:03.58,Default,,0000,0000,0000,,That's called a "period-doubling cascade"\Nfor the obvious reason.
Dialogue: 0,0:03:03.80,0:03:08.04,Default,,0000,0000,0000,,I also showed you in the last segment\Nthat there were regions of order
Dialogue: 0,0:03:08.16,0:03:12.30,Default,,0000,0000,0000,,within the chaos; that is, for some\NR-value, there was chaos,
Dialogue: 0,0:03:12.47,0:03:16.36,Default,,0000,0000,0000,,but then if you raised R a little bit, you\Nwent back into a periodic regime.
Dialogue: 0,0:03:16.36,0:03:22.16,Default,,0000,0000,0000,,This particular periodic regime starts out\Nwith a 3-cycle, and then goes to a 6-cycle
Dialogue: 0,0:03:22.19,0:03:24.29,Default,,0000,0000,0000,,and a 12-cycle, and so on and so forth.
Dialogue: 0,0:03:24.39,0:03:27.85,Default,,0000,0000,0000,,So it's another period-doubling\Nbifurcation sequence.
Dialogue: 0,0:03:28.63,0:03:31.74,Default,,0000,0000,0000,,You may remember, in the very first \Nsegment of this course,
Dialogue: 0,0:03:31.82,0:03:35.52,Default,,0000,0000,0000,,I showed you the title page of a paper \Ncalled "Period-3 Implies Chaos".
Dialogue: 0,0:03:35.85,0:03:42.27,Default,,0000,0000,0000,,The fact there is a period-3 orbit\Nin this map is very, very significant.
Dialogue: 0,0:03:42.41,0:03:47.21,Default,,0000,0000,0000,,And if people are interested in that, I\Ncan record an auxiliary video about that.
Dialogue: 0,0:03:47.58,0:03:50.28,Default,,0000,0000,0000,,Another interesting thing to note about\Nthis structure
Dialogue: 0,0:03:50.49,0:03:54.16,Default,,0000,0000,0000,,is that it contains small \Ncopies of itself.
Dialogue: 0,0:03:54.16,0:03:58.58,Default,,0000,0000,0000,,If you were to zoom in on that piece of\Nthe structure inside the red circle,
Dialogue: 0,0:03:59.00,0:04:01.05,Default,,0000,0000,0000,,it would look like the whole structure.
Dialogue: 0,0:04:01.11,0:04:03.40,Default,,0000,0000,0000,,That is, this is a fractal object.
Dialogue: 0,0:04:03.97,0:04:06.58,Default,,0000,0000,0000,,I'm sure many of you have heard about fractals.
Dialogue: 0,0:04:07.21,0:04:11.06,Default,,0000,0000,0000,,Fractals are sets that have non-integer\NHausdorff dimension
Dialogue: 0,0:04:11.15,0:04:13.01,Default,,0000,0000,0000,,(mathematically, that's the formal term).
Dialogue: 0,0:04:13.20,0:04:14.96,Default,,0000,0000,0000,,Informally, they're "self-similar".
Dialogue: 0,0:04:15.08,0:04:19.06,Default,,0000,0000,0000,,The second row of images here show\Nsomething called the Koch curve.
Dialogue: 0,0:04:19.25,0:04:22.77,Default,,0000,0000,0000,,The way you construct this fractal is by\Ntaking an equilateral triangle,
Dialogue: 0,0:04:22.93,0:04:28.22,Default,,0000,0000,0000,,and then taking 3 equilateral triangles,\N1/3 the size in the sense of edge-length,
Dialogue: 0,0:04:28.42,0:04:30.96,Default,,0000,0000,0000,,and sticking them to each exposed face\Nof that thing.
Dialogue: 0,0:04:31.23,0:04:33.33,Default,,0000,0000,0000,,Then you iterate; you take little \Ntriangles
Dialogue: 0,0:04:33.38,0:04:37.16,Default,,0000,0000,0000,,and stick them to the sides of each of \Nthose pointy faces, and keep going.
Dialogue: 0,0:04:37.41,0:04:41.29,Default,,0000,0000,0000,,Eventually, you'll get this beautiful \Nstructure that looks alot like a snowflake
Dialogue: 0,0:04:41.56,0:04:43.91,Default,,0000,0000,0000,,Fractals play an interesting role \Nin mathematics.
Dialogue: 0,0:04:44.19,0:04:48.29,Default,,0000,0000,0000,,There are also lots of examples of frac- \Ntals & fractal-like structures in nature.
Dialogue: 0,0:04:48.36,0:04:49.67,Default,,0000,0000,0000,,Here's an example.
Dialogue: 0,0:04:50.94,0:04:54.64,Default,,0000,0000,0000,,Fractals are also useful analogs\Nfor nature in computer graphics.
Dialogue: 0,0:04:55.16,0:04:57.97,Default,,0000,0000,0000,,Here's a beautiful fractal called\Nthe Mandelbrot set,
Dialogue: 0,0:04:58.02,0:05:01.48,Default,,0000,0000,0000,,and this video is showing you that if you\Nzoom in on the Mandelbrot set,
Dialogue: 0,0:05:01.57,0:05:03.47,Default,,0000,0000,0000,,you keep seeing more and more structure;\N
Dialogue: 0,0:05:03.58,0:05:06.63,Default,,0000,0000,0000,,in fact, you keep seeing structure that is\Nself-similar.
Dialogue: 0,0:05:10.99,0:05:15.22,Default,,0000,0000,0000,,There's a whole new Mandelbrot set \Nway down in the tendrils of the old one.
Dialogue: 0,0:05:15.22,0:05:17.36,Default,,0000,0000,0000,,And you can keep zooming in \Nand zooming in,
Dialogue: 0,0:05:17.36,0:05:19.69,Default,,0000,0000,0000,,and you'll keep seeing \Nself-similar structure.
Dialogue: 0,0:05:20.47,0:05:25.18,Default,,0000,0000,0000,,I've included a link to that video on the\Nsupplementary materials section
Dialogue: 0,0:05:25.25,0:05:27.80,Default,,0000,0000,0000,,of the Complexity Explorer website\Nfor this course,
Dialogue: 0,0:05:28.40,0:05:33.03,Default,,0000,0000,0000,,right here, under the section for this\Nsegment of this unit.
Dialogue: 0,0:05:33.43,0:05:35.01,Default,,0000,0000,0000,,Remember, this is where you should go\N
Dialogue: 0,0:05:35.54,0:05:38.57,Default,,0000,0000,0000,,for links to materials that you might need\Nto do the homework,
Dialogue: 0,0:05:38.88,0:05:40.83,Default,,0000,0000,0000,,like this Logistic Map app,
Dialogue: 0,0:05:41.15,0:05:44.56,Default,,0000,0000,0000,,for materials like this paper, which you \Nwould look at
Dialogue: 0,0:05:44.63,0:05:48.20,Default,,0000,0000,0000,,if you wanted to learn more about the\Nconcepts that I talked about
Dialogue: 0,0:05:48.22,0:05:49.76,Default,,0000,0000,0000,,in that segment.
Dialogue: 0,0:05:49.76,0:05:53.90,Default,,0000,0000,0000,,And I've also included some links to \Ntutorial materials
Dialogue: 0,0:05:53.93,0:05:58.30,Default,,0000,0000,0000,,and other sorts of things that might help\Nyou if you need some background to fill in
Dialogue: 0,0:05:59.21,0:06:02.76,Default,,0000,0000,0000,,And here's an important thing: the \Nconnection between fractals and chaos.
Dialogue: 0,0:06:03.24,0:06:06.46,Default,,0000,0000,0000,,There is a connection, but it is not an\N"if-and-only-if".
Dialogue: 0,0:06:06.80,0:06:09.94,Default,,0000,0000,0000,,Many chaotic systems have some\Nfractal structure,
Dialogue: 0,0:06:10.02,0:06:14.04,Default,,0000,0000,0000,,but it is by no means the case that all\Nchaotic systems have fractal structure;
Dialogue: 0,0:06:14.34,0:06:17.74,Default,,0000,0000,0000,,that is, there are chaotic systems that \Ndo not have fractal structure,
Dialogue: 0,0:06:17.86,0:06:21.00,Default,,0000,0000,0000,,there are certainly tons of fractals\Nthat have nothing to do with chaos,
Dialogue: 0,0:06:21.12,0:06:24.66,Default,,0000,0000,0000,,but the popular science press has\Nconflated these two topics.
Dialogue: 0,0:06:25.20,0:06:27.19,Default,,0000,0000,0000,,If you want to learn more about fractals,
Dialogue: 0,0:06:27.26,0:06:31.63,Default,,0000,0000,0000,,you can take a look at Dave Feldman's \Ncourse on the Complexity Explorer MOOC.
Dialogue: 0,0:06:31.85,0:06:34.62,Default,,0000,0000,0000,,One last point here, relating to \Ntransient length:
Dialogue: 0,0:06:34.72,0:06:38.52,Default,,0000,0000,0000,,remember that for some R-values,\Nthe transient was really long?
Dialogue: 0,0:06:38.65,0:06:41.73,Default,,0000,0000,0000,,How do you think that will manifest\Nin a bifurcation diagram?
Dialogue: 0,0:06:42.05,0:06:45.11,Default,,0000,0000,0000,,That is, there is some fixed point here,\Nbut the trajectory is taking
Dialogue: 0,0:06:45.15,0:06:46.81,Default,,0000,0000,0000,,a really long time to get there.
Dialogue: 0,0:06:46.89,0:06:50.42,Default,,0000,0000,0000,,What that will look like on a slice of the\Nbifurcation diagram is this.
Dialogue: 0,0:06:51.79,0:06:55.81,Default,,0000,0000,0000,,That's hard to see, but I'm trying to draw\Na series of points coming up from the axis
Dialogue: 0,0:06:55.95,0:06:59.82,Default,,0000,0000,0000,,and slowly getting closer and closer and\Ncloser, but taking forever to get there.
Dialogue: 0,0:07:00.26,0:07:02.43,Default,,0000,0000,0000,,So if we want to see the asymptotic \Nbehavior,
Dialogue: 0,0:07:02.43,0:07:06.49,Default,,0000,0000,0000,,we want to throw out the transient, but\Nhow many points do we need to throw out
Dialogue: 0,0:07:06.58,0:07:08.87,Default,,0000,0000,0000,,if we want to get rid \Nof the transient here?
Dialogue: 0,0:07:09.96,0:07:13.72,Default,,0000,0000,0000,,To get rid of the transient, we actually\Nneed another step in our code here.
Dialogue: 0,0:07:13.82,0:07:16.83,Default,,0000,0000,0000,,Really what we need to do is iterate a \Nwhole bunch of times,
Dialogue: 0,0:07:16.86,0:07:18.85,Default,,0000,0000,0000,,but not plot those points,
Dialogue: 0,0:07:18.85,0:07:21.02,Default,,0000,0000,0000,,and then from the ending point of \Nthat orbit,
Dialogue: 0,0:07:21.02,0:07:24.36,Default,,0000,0000,0000,,iterate a bunch more times, and plot \Nthose points.
Dialogue: 0,0:07:24.42,0:07:27.33,Default,,0000,0000,0000,,That amounts to omitting the transient.
Dialogue: 0,0:07:27.33,0:07:31.24,Default,,0000,0000,0000,,But the question is, these words: how do \Nyou pick how many points to iterate
Dialogue: 0,0:07:31.34,0:07:34.78,Default,,0000,0000,0000,,to get rid of the transient, and how do\Nyou pick how many points to plot
Dialogue: 0,0:07:34.79,0:07:37.38,Default,,0000,0000,0000,,so that you get a really nice picture?\NThose are both tricky.
Dialogue: 0,0:07:37.38,0:07:42.30,Default,,0000,0000,0000,,You want the red bunch number to be \Nlarge enough so that you see the structure
Dialogue: 0,0:07:42.51,0:07:47.95,Default,,0000,0000,0000,,but not so large that the finite size of \Nthe plotted points obscures the structure.
Dialogue: 0,0:07:48.32,0:07:51.98,Default,,0000,0000,0000,,And you want to throw out enough points\Nso the transient has really died out,
Dialogue: 0,0:07:52.03,0:07:54.57,Default,,0000,0000,0000,,but how long is that? There's no way to \Nknow, really.
Dialogue: 0,0:07:54.84,0:07:58.17,Default,,0000,0000,0000,,And they tend to get longer just before\Na bifurcation.
Dialogue: 0,0:07:58.58,0:08:02.92,Default,,0000,0000,0000,,In practice, what you do is increase the\Nnumber of points that you throw away
Dialogue: 0,0:08:02.92,0:08:06.98,Default,,0000,0000,0000,,before plotting, until the periodic orbits\Nare crisp on your plots.
Dialogue: 0,0:08:07.05,0:08:11.35,Default,,0000,0000,0000,,That amount of thrown-away points is \Noverkill far away from the bifurcations,
Dialogue: 0,0:08:11.35,0:08:13.49,Default,,0000,0000,0000,,of course, where the transient is short,
Dialogue: 0,0:08:13.49,0:08:16.84,Default,,0000,0000,0000,,but otherwise, your orbits will thicken \Nup near the bifurcation point.
Dialogue: 0,0:08:17.27,0:08:21.55,Default,,0000,0000,0000,,All of that will play a role in the next\Nsegment, where we'll dig into the pattern
Dialogue: 0,0:08:21.55,0:08:26.33,Default,,0000,0000,0000,,behind the shrinking widths and heights \Nof the pitchforks in the bifurcation plot.